I have been unsuccessful in finding the definitive year via the web, because of all
the conflicting citations. The next step is to retrieve
Mathematische Annalen volume 55, but perhaps someone can spare me that trouble...?
Thanks!

The paper can be viewed though the Springer webpages springerlink.com/content/h54v26316705n28r. The 55. volume of Math. Ann. has papers both from 1901 and 1902, which perhaps explains the confusion. Dehn's paper is in the number 3. issue, which is listed as 'September 1901'.
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J.C. OttemAug 25 '11 at 19:13

Thanks to J.C., Andreas, and Carlo, all of whom explained the mystery!
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Joseph O'RourkeAug 25 '11 at 19:35

according to MathSciNet and Springer confirms this here. But on the scanned original provided by the Göttingen Center for Digitalisation here, the bottom line of the first page shows 1902. However, this refers to all issues of Volume 55. Looking more closely, only issues 1-3 were published in 1901 and issue 4 was published in 1902.

I guess, this means that MathSciNet is correct. However, the confusion is somewhat understandable.

Another point to consider is whether "Über den Rauminhalt"
is in fact Dehn's first solution to Hilbert's 3rd Problem. I
believe his first solution was in the paper "Über raumgleiche
Polyeder" in the Nachrichten der Königliche Gesellschaft der
Wissenschaften zu Göttingen of 1900, pp. 345 -- 354.

Yes, Hilbert refers to that in his own paper. Hilbert calls this a "note" and the other a "paper," which I took to suggest that the former was a type of preliminary version of the latter.
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Joseph O'RourkeAug 25 '11 at 23:04

Just for the record, here is Hilbert's footnote at the end of his 3rd problem description: "Since this was written Herr Dehn has succeeded in proving this impossibility. See his note: "Über raumgleiche Polyeder," in Nachrichten d. K. Gesellsch. d. Wiss. zu Göttingen, 1900, and a paper soon to appear in the Math. Annalen [vol. 55, pp. 465-478]."
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Joseph O'RourkeAug 26 '11 at 22:06